Our mathematical research focus on linear and nonlinear partial differential equations (PDEs), functional analysis, and operator theory, and here mostly operator theoretic methods in differential equations, motivated by problems in quantum physics. Analogous to Newton’s equations, describing the motion of macroscopic bodies, from falling apples to space crafts orbiting the Earth, the Schrödinger equation describes microscopic phenomena, from the simplest atoms to processes inside stars. The differential operator appearing in the Schrödinger equation, the so-called Schrödinger operator, characterizes the physical system and it is therefore one of the most interesting objects in mathematical physics; its spectral theory has deep roots in non-relativistic quantum mechanics. A number of mathematical notions and theories were inspired by the needs of the theory of the Schrödinger operator and its generalizations.
Here are some themes and techniques that we currently work on:
Resonances in Quantum Chemistry. Schrödinger’s theory of quantum mechanics distinguishes bound states and scattering states, the former being associated with point spectrum and the latter with the continuous spectrum. These two states are linked by resonances associated with meta-stable (or quasi-stationary) states for certain types of quantum systems and since the beginning of quantum theory such resonances have posed some of the most notoriously difficult challenges. Resonances are pseudo-eigenvalues of certain classes of Schrödinger operators and the pseudo-eigenvalues are characterized by having nonzero imaginary parts which determine the decay rates of the corresponding unstable states. A priori, this sounds as a contradiction because a Schrödinger operator H acting on a Hilbert space is self-adjoint and, as a consequence, its spectrum is real. For this reason, some mathematicians think that the foundations of resonance theory is still not on a firm ground. However, a resonance z is associated with a “resonance eigenfunction” satisfying the equation H f = z f where the function f satisfies a condition at infinity that implies that it does not belong to the afore-mentioned Hilbert space. This approach has lead to the computation of resonances that correspond to quantities that can be measured experimentally, and the understanding that any definition of resonance must depend not only on the Schrödinger operator itself, but also on some additional background data.
Mathematical analysis of Density Functional Theory in internal and external magnetic fields
Phase space bounds and eigenvalue asymptotics for Schrödinger operators.
Spectral and scattering properties near thresholds for scalar-valued and matrix-valued Schrödinger operators.
Spectral and scattering properties for Schrödinger and Dirac operators with a constant magnetic field.
Spectral and scattering properties of quantum wires.